Asael or Asahel Smith was born in Topsfield, MA, in 1744 and married Mary Duty or Doty there on 12 Feb 1767. Their first three children (Jesse, Priscilla and Joseph) were born in Massachusetts prior to the Smith family moving to New Hampshire, where the remainder of their eleven children were born. Both Mary (1775) and Samuel (1777) were born in Windham, Rockingham County, NH.
For the first time online, Footnote makes available pages of the Revolutionary War Rolls indicating that an Asael/Asail Smith served in Regiments of New Hampshire. One document is dated 9 July 1776 and the other is from 10 Aug 1776. You can see them to the right. His name is at the bottom of the column on the right-hand side on both pages.